Proof. A Borel \({\mathbb {Z}}^d\) -action defines a hyperfinite equivalence relation (see [1]}, [2]}). Since \(\Gamma \)  is a \(\mathbb {Z}^2\) -action, \(E_{X}(\Gamma )\)  is hyperfinite, hence generated by a Borel \(\mathbb {Z}\) -action (see Theorem \(\ref {thm hyperfinite}\) ). Thus there exists \(R \in Aut(X,{\mathcal {B}})\) , such that for every \(x \in X\) , \(\Gamma x = \lbrace R^i x : i \in {\mathbb {Z}}\rbrace \) .
